Combining conditional heteroscedasticity and extreme value theory to measure asset risks

Abstract

One approach to financial modelling is that of applying conditional heteroscedastic models such as ARCH/GARCH models to explain the error dynamics of models fitted to financial time series. Extensions to ARCH and GARCH models have also been proposed but we shall not be going into these in this study. Extreme value theory (EVT) is an on going approach to financial modelling. This type of modelling is more appropriate at catering for the occurrence of extreme values, but is a distributional approach and hence does not cater for the dependency between observations. In this study we seek to combine conditional heteroscedastic modelling with extreme value theory. The idea here is that of assuming that standardized errors may be modelled by an extreme value distribution, hence combining the benefits of the two respective approaches: the ability to cater for serial correlation in the errors in the former, and the ability to model the occurrence of extreme values in the latter. The technique we shall use here is one by Mc Neil and Frey (2002), which combines ARCH/GARCH modelling with the peak over threshold (POT) method in extreme value theory. We shall obtain the one-step ahead value at risk and expected shortfall for the above model and compare it with similar measures obtained when the standardized errors are assumed to be normal and t-distributed.